Evaluate lim_x→2 (x³ - 3x² + 2) / (x-2) using l’Hôpital’s Rule and then check your work by evaluating the limit using an appropriate Chapter 2 method.

Verified step by step guidance

1. First, identify that the limit lim_x→2 (x³ - 3x² + 2) / (x-2) is an indeterminate form of type 0/0, which is suitable for l'Hôpital's Rule.
2. Apply l'Hôpital's Rule, which states that if lim_x→c f(x)/g(x) is of the form 0/0 or ∞/∞, then lim_x→c f(x)/g(x) = lim_x→c f'(x)/g'(x), provided the limit on the right exists.
3. Differentiate the numerator f(x) = x³ - 3x² + 2 to get f'(x) = 3x² - 6x.
4. Differentiate the denominator g(x) = x - 2 to get g'(x) = 1.
5. Evaluate the limit lim_x→2 (3x² - 6x) / 1, which simplifies to lim_x→2 (3x² - 6x). Now, check your work by factoring the original expression: (x³ - 3x² + 2) = (x-2)(x² - x - 1) and cancel the (x-2) terms, then evaluate lim_x→2 (x² - x - 1).

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a method for evaluating limits of indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) results in an indeterminate form as x approaches a value, then the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately, and then re-evaluating the limit.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. In the given limit, the expression x³ - 3x² + 2 is a polynomial, and understanding its behavior as x approaches a specific value is crucial for limit evaluation, especially when factoring or simplifying the expression.

Limit Evaluation Techniques

Limit evaluation techniques include various methods for finding the limit of a function as it approaches a certain point. These methods can involve direct substitution, factoring, rationalizing, or using L'Hôpital's Rule. In this case, after applying L'Hôpital's Rule, one can also check the limit by substituting the value directly into the simplified expression or by using polynomial long division.